Calculator
Use consistent units. If you enter more than one timing input, angular frequency is used first, then period, then frequency.
Example data table
These sample values show how harmonic inputs map to displacement outputs.
| Case | Wave | A | T | φ | t | d | x |
|---|---|---|---|---|---|---|---|
| 1 | Sine | 5 m | 2 s | 0° | 0.25 s | 0 m | 3.5355 m |
| 2 | Cosine | 10 cm | 4 s | 90° | 0.5 s | 2 cm | -5.0711 cm |
| 3 | Sine | 3 mm | 1 s | 30° | 0.1 s | -1 mm | 1.7406 mm |
| 4 | Cosine | 12 m | 6 s | 0° | 1.2 s | 0 m | 3.7082 m |
Formula used
Displacement equation: x = d + A sin(ωt + φ) or x = d + A cos(ωt + φ)
Meaning: x is displacement, d is offset, A is amplitude, ω is angular frequency, t is time, and φ is the phase constant.
Period and frequency conversions: ω = 2π / T, f = 1 / T, and ω = 2πf
Interpretation: these three values describe the same timing behavior in different forms.
Velocity and acceleration: for sine motion, v = Aω cos(ωt + φ). For cosine motion, v = -Aω sin(ωt + φ). In both cases, a = -ω²(x - d).
Why it helps: these outputs show motion direction and restoring behavior, not only position.
Inverse timing solve: when solving period, frequency, or angular frequency, the calculator first finds a valid phase angle from the displacement ratio r = (x - d)/A.
Then: ω = (θ - φ + 2πn) / t, where n is the cycle index. This is why multiple solutions may exist.
How to use this calculator
- Choose what you want to solve for.
- Select sine or cosine motion.
- Enter available motion values in consistent units.
- Set phase in degrees or radians.
- For inverse timing solves, choose a branch and cycle index.
- Press Calculate to show the result above the form.
- Review the graph, derived outputs, and history table.
- Use the CSV or PDF buttons to export your report.
FAQs
1) What does amplitude mean in harmonic motion?
Amplitude is the greatest distance from equilibrium. It describes the motion size, not the speed. Larger amplitude usually means larger displacement, higher peak speed, and higher peak acceleration when timing stays fixed.
2) What is the period of oscillation?
The period is the time for one full cycle. After one period, the object repeats the same position, velocity direction, and phase state. Frequency is simply the reciprocal of the period.
3) How is displacement different from amplitude?
Displacement is the instantaneous position at a chosen time. Amplitude is the maximum possible distance from equilibrium. Displacement changes continuously, while amplitude stays fixed for an ideal undamped oscillation.
4) Can I use degrees instead of radians?
Yes. Select degrees in the phase unit field. The calculator converts the phase internally before applying the harmonic formulas, so the final answer remains consistent with the motion equation.
5) Why do inverse period or frequency solves need branch selection?
A sine or cosine wave reaches the same displacement at more than one phase angle. Branch selection chooses one inverse angle, while the cycle index adds full turns to capture repeating solutions.
6) Why can amplitude solving fail at some times?
If the sine or cosine factor becomes zero, the displacement equation loses direct amplitude information at that instant. Changing time, phase, or wave type usually removes that zero-factor condition.
7) Does the graph use the solved values?
Yes. The Plotly graph is built from the current solved motion model. It traces displacement across time and marks the selected time point, so you can visually confirm the reported displacement.
8) Which units should I enter?
Use any consistent unit system. For example, meters with seconds or centimeters with milliseconds. The calculator does not force SI, but inconsistent units will produce misleading physical interpretations.